Short answers, pulled from the story.
The United Nations had 51 original founding members. Fifty of them signed the UN Charter at the United Nations Conference on International Organization in San Francisco on the 26th of June 1945, while Poland, which was not represented at the conference, signed on the 15th of October 1945. The UN officially came into existence on the 24th of October 1945.
A new member must first receive affirmative votes from at least nine of the fifteen members of the UN Security Council, with none of the five permanent members using their veto. The Security Council's recommendation is then approved by a two-thirds majority vote in the General Assembly. Only sovereign states are eligible for membership.
The 1955 package deal was a simultaneous admission of sixteen countries that had been blocked from joining the UN due to Cold War vetoes between the United States and the Soviet Union. On the 14th of December 1955, under Security Council Resolution 109, Albania, Jordan, Ireland, Portugal, Italy, Austria, Finland, Ceylon, Nepal, Libya, Cambodia, Laos, Hungary, Romania, Bulgaria, and Spain were all admitted at once.
United Nations General Assembly Resolution 2758, passed on the 25th of October 1971, recognized the People's Republic of China as the only lawful representative of China and expelled the representatives of Chiang Kai-shek. The resolution transferred the China seat, including the permanent Security Council seat, from the Republic of China to the People's Republic of China.
No member state has ever been formally expelled under Articles 5 or 6 of the UN Charter. In October 1974, a resolution to expel South Africa over apartheid was vetoed by France, the United Kingdom, and the United States. The General Assembly suspended South Africa from its 29th session in November 1974, but South Africa was welcomed back after its democratic elections on the 23rd of June 1994.
Indonesia is the only member state to have attempted withdrawal. In a letter dated the 20th of January 1965, Indonesia informed the Secretary-General of its decision to withdraw during the Indonesia-Malaysia confrontation. After President Sukarno was overthrown, Indonesia reversed course in September 1966, and the UN treated the episode as a temporary cessation of cooperation rather than a formal withdrawal, readmitting Indonesia without a new membership application.